Quadruples, admissible elements and Herrmann's endomorphisms

Details Quadruples, admissible elements and Herrmann's endomorphisms Quadruples, admissible elements and Herrmann's endomorphisms

2006-05-26

arxiv.org/abs/math/0605672v1

J. Pure Appl. Algebra 211 (2007), no. 1, 95--202

Mathematics

Representation Theory

44 pages, 7 figures

Scientific paper

10.1016/j.jpaa.2007.01.005

We obtain a connection between admissible elements for quadruples and Herrmann's endomorphisms. Herrmann constructed perfect elements $s_n$, $t_n$, $p_{i,n}$ in $D^4$ by means of some endomorphisms and showed that these perfect elements coincide with the Gelfand-Ponomarev perfect elements modulo linear equivalence. We show that the admissible elements in $D^4$ are also obtained by means of Herrmann's endomorphisms $\gamma_{ij}$. Endomorphism $\gamma_{ij}$ and the elementary map of Gelfand-Ponomarev $\phi_i$ act, in a sense, in opposite directions, namely the endomorphism $\gamma_{ij}$ adds the index to the start of the admissible sequence, and the elementary map $\phi_i$ adds the index to the end of the admissible sequence.

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